The cheapest hostel in Reykjavik is the official HI hostel but it is a bit away from the city centre and it might not be as nice as a lot of the downtown hostels. Try to eat out as little as you can and buy food in supermarkets. Bónus is most often the cheapest. The HI hostel has a kitchen you can use. PM me if you have any more questions and I will try to answer them.